Overview
This lecture focuses on the practical steps to construct and interpret confidence intervals for real-life applications, using a population proportion example and emphasizing calculator use.
The Three Steps for Confidence Intervals
- Step 1: Check if the sample is valid by verifying the Central Limit Theorem conditions.
- Step 2: Use a graphing calculator (TI84 or 83) to compute the confidence interval.
- Step 3: Interpret the confidence interval in a complete sentence using a provided template.
Step 1: Checking Conditions (Central Limit Theorem)
- Condition 1: Confirm the sample was collected randomly (look for "random" in the prompt).
- Condition 2: Ensure the sample is large enough by checking both the number of successes (yes responses) and failures (no responses).
- Success in this problem: Respondents willing to pay higher prices (518).
- Failure: Respondents not willing to pay more (1,154 - 518 = 636).
- Both the number of successes and failures must be greater than 10, which they are in this case.
- Condition 3: Population size must be at least 10 times the sample size (here, more than 11,540 adult Americans, which is true).
Example Problem Recap
- Survey: 1,154 adults asked if willing to pay higher prices to protect the environment.
- 518 said yes (successes), 636 said no (failures).
- All three Central Limit Theorem conditions are satisfied, confirming a good sample.
Structure of Confidence Interval Problems
- Exam and homework questions provide practical prompts (in black).
- Scaffolding or problem outlines/templates (in purple) are given to ensure stepwise completion.
- The template for interpretation remains the same for every confidence interval problem.
Key Terms & Definitions
- Confidence Interval — A range of values, derived from the sample, used to estimate a population parameter.
- Central Limit Theorem — A statistical theory stating sample means will approximate a normal distribution if the sample is large and chosen randomly.
- Success — The number of subjects with the attribute of interest (e.g., yes responses).
- Failure — The number of subjects without the attribute of interest (e.g., no responses).
Action Items / Next Steps
- Practice constructing confidence intervals using your graphing calculator.
- Use the provided purple template for interpreting confidence intervals in homework.
- Review Section 7.3 on verifying sample conditions if needed.